Enumeration of Corners in Tree-like Tableaux

TL;DR

This paper confirms conjectures on counting corners in various tableaux types, providing new enumeration methods and bijections, and explores polynomial analogues with implications for the PASEP model.

Contribution

It introduces new enumeration formulas for corners in tree-like tableaux and symmetric variants, using bijections to permutations, and proposes polynomial analogues related to the PASEP.

Findings

Confirmed conjectures on corner enumeration in tree-like tableaux

Established bijections linking corners to permutation statistics

Proposed polynomial analogues with implications for PASEP

Abstract

In this paper, we confirm conjectures of Laborde-Zubieta on the enumeration of corners in tree-like tableaux and in symmetric tree-like tableaux. In the process, we also enumerate corners in (type $B$) permutation tableaux and (symmetric) alternative tableaux. The proof is based on Corteel and Nadeau's bijection between permutation tableaux and permutations. It allows us to interpret the number of corners as a statistic over permutations that is easier to count. The type $B$ case uses the bijection of Corteel and Kim between type $B$ permutation tableaux and signed permutations. Moreover, we give a bijection between corners and runs of size 1 in permutations, which gives an alternative proof of the enumeration of corners. Finally, we introduce conjectural polynomial analogues of these enumerations, and explain the implications on the PASEP.